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Publication Partner:
International Journal of Scientific and Research Publications (ISSN: 2250-3153)
1
2021
Authored by:
Dr. Sarika D Patil
Dr. Sumant G Kadwane
Mr.Akshay D Kadu
IJSRP INC.
Hybrid Optimization Approach for Harmonic
Mitigation in Multilevel Inverter
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Hybrid Optimization Approach
for Harmonic Mitigation in
Multilevel Inverter
Dr. Sarika D Patil
Dr. Sumant G Kadwane
Mr.Akshay D Kadu
Publishing Partner:
IJSRP Inc.
www.ijsrp.org
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Preface
Power electronics being a energy efficient devices have major role in power conversion
application systems. To guaranteed power quality the significant integration of renewable energy
is must in power system. Generally for medium and high power applications multilevel inverters
are widely used. But due to harmonics present in the system, the power quality is disturbed
usually. In multilevel inverters, switching techniques improve the harmonic profile and ensures
its power quality. Mostly selective harmonic elimination techniques are used for lowering
harmonics and total harmonic distortion also. Selective harmonic elimination method is a
fundamental switching frequency method having lower switching losses and high efficiency. The
output obtained from multilevel inverter consists of nonlinear transcendental equations and need
specific optimization algorithms. The solution to these equations provides optimized switching
angles which further reduces the specific harmonics along with THD for the system. In this
monograph, a hybrid optimization algorithm is used for solving these nonlinear transcendental
equations and obtaining the desired solution. This hybrid algorithm is a grouping of N-R
algorithm with Ant colony algorithm. This algorithm is used whenever there are uncertainties in
the system model. This hybrid optimization algorithm proved as a successful algorithm with the
help of simulated results.
I precisely mention my sincere thanks for all the members who directly or indirectly
helped me, guided me and encouraged me to complete this monograph.
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Copyright and Trademarks
All the mentioned authors are the owner of this Monograph and own all copyrights of the Work.
IJSRP acts as publishing partner and authors will remain owner of the content.
Copyright©2021, All Rights Reserved
No part of this Monograph may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise,
except as described below, without the permission in writing of the Authors& publisher.
Copying of content is not permitted except for personal and internal use, to the extent permitted
by national copyright law, or under the terms of a license issued by the national Reproduction
Rights Organization.
Trademarks used in this monograph are the property of respective owner and either IJSRP or
authors do not endorse any of the trademarks used.
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Authors
Dr. Sarika D Patil Assistant Professor,
Department of Electrical Engineering
Yeshwantrao Chavan College of Engineering, Nagpur
Nagpur -441110
Dr. Sumant G Kadwane Professor,
Department of Electrical Engineering
Yeshwantrao Chavan College of Engineering, Nagpur
Nagpur -441110
Mr. Akshay D Kadu Assistant Professor,
Department of Electrical Engineering
Yeshwantrao Chavan College of Engineering, Nagpur
Nagpur -441110
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Table of Content
1. INTRODUCTION 7
2. MODULATION TECHNIQUES 8
2.1 TOPOLOGIES RELATED TO MULTILEVEL INVERTER 8
2.2 MODULATION TECHNIQUES 8
3. SELECTIVE HARMONIC ELIMINATION 10
3.1 NEWTON RAPHSON ALGORITHM 12
4. HYBRID OPTIMIZATION ALGORITHM 16
5. CONCLUSION 19
6. FUTURE SCOPE 20
7. REFERENCES 21
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1. INTRODUCTION
he Multilevel inverters are generally preferred where medium to high voltage drives like electric vehicles, grid-
connected systems etc are used. It is widely used for the purpose of increasing power ratings of the devices,
improvement for harmonic profile, and reduction for electromagnetic interference. All these advantages can be
accomplished with use of several DC voltage levels to obtain output voltage waveform.
In the multilevel inverters, all harmonics cannot be removed with the help of filters. Only few higher order
harmonics are removed by the appropriate choice of passive filters. Whereas the lower order harmonics cannot be
removed with the help of passive filters. Also when the frequency increases the losses will also increase for high
voltages. Hence through optimizations technique, this objective can be achieved which proves the recent
contribution of multilevel inverter technology.
So many approaches have made for solving nonlinear equations and getting its approximate solution upto date
and again research is going on. The first approach was obtaining solution by resultant theory approach. From this
method desired solution sets are achieved by converting nonlinear equations into polynomial sets and then by
applying resultant theory. This method was successful to some extent but it was a time consuming algorithm. Also
when DC level changes it requires new expressions to solve. To overcome these drwabacks iterative methods came
into existence. In iterative methods like Newton Raphson and Gauss Siedal methods, solution to nonlinear
transcendental equations can be obtained. But in Newton Raphson algorithm, initial guess closer to approximate
solution is required otherwise it may take time to find the possible solution sets and the system may take longer time
to converge. In Gauss Siedal method, after successful iterations, it provides solution to nonlinear equations for
obtaining desired values but discontinuities in the solution may occur for non-existing solutions.
So, owing to limitations and drawbacks of conventional iterative methods, next approach was based on
stochastic search optimization approach. These approaches were mainly nature inspired population-based
algorithms. These algorithms include GA, PSO Algorithm, Firefly Algorithm, Bee Algorithm, BBO Algorithm and
many more. These algorithms are again proved to be successful for solving nonlinear equations and getting desired
solutions. But again these algorithms had some drawbacks and limitations. These algorithms can be implemented to
symmetrical design of multilevel inverter with equivalent DC input sources. Secondly, all solution sets cannot be
obtained for all modulation indices. Also it requires more computational time. As the number of levels in multilevel
inverter increases, complexity in the design of system is also increases.
Other optimization algorithms like Ant Colony Optimization (ACO) were invented by Researcher Marco
Dorigo for solving unique traveling salesman problem. Ant Colony algorithm developed applied and verified
successfully on typical problems, like battery charging problems, vehicle navigation problems, unit commitment,
etc. ACO algorithms have assured convergence but time for convergence may be more depending upon the system
parameters and constraints.
From the above discussion, it is obvious that main challenge is to obtain desired solutions for nonlinear
transcendental equations derived from fourier series expansion of output of multilevel inverter. Owing to the above
drawbacks and limitations of all evolutionary algorithms, need of new improved strong algorithm is to be explored
which will maintain all concerns related to selective harmonic elimination issues. In this monograph, a new
proposed hybrid algorithm is implemented including combination of optimization algorithm i.e. ACO algorithm and
iterative algorithm i.e. Newton Raphson Algorithm. This algorithm proves to be very successful algorithm with less
computational burden and faster convergence rate.
T
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2. MODULATION TECHNIQES
2.1 Topologies related to Multilevel Inverter
Generally multilevel inverters are investigated for the purpose of reduction of size as well as price of filters
with less harmonic disturbance. VSC i.e. Voltage Source multilevel inverters are very much popular now a
days because of its applications in motor drives, UPS systems and FACTS Devices. In multilevel inverter
desired output can be obtained with any level using dissimilar DC inputs. This is the most important
characteristics of multilevel inverter. For attaining this, the different topologies of multilevel inverter are
neutral-point clamped, flying capacitor, cascaded H-Bridge inverter along with different dc levels and
modular structures have been proposed by many researchers till date. Figure 2.1 shows the different
topologies of inverter.
Neutral Clamped
MLI
Cascaded H-Bridge
MLICapacitor Clampled
MLI
Modular MLI
Multilevel Inverters
Fig. 2.1: Topologies of MLI
2.2 Modulation Techniques
In multilevel inverter power devices are mainly used. But the drawback of these power devices is-high switching
losses and reduced efficiency and thereby reducing the life of the system. This drawback can be overcome by
selecting fundamental switching frequency for switching the inverter. It will ensure the less load current
disturbances and improvement in the performance dynamically. Hence anew pulse width modulation technique with
high performance is introduced to overcome this issue.
Numerous modulation techniques have been suggested for MLI. These techniques are mainly classified as
fundamental switching techniques, mixed switching techniques also high switching frequency switching techniques
given in Figure 2.2.
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Modulation
Techniques
Fundamental
Switching
Technique
Mixed
Switching
Technique
High Switching
Technique
Space
vector
Selective
Harmonic
Elemination
Hybrid
Multilevel
Modulation
Space
Vector
PWM
Phase
Shifted PWM
Line
Shifted
PWM
Phase
Disposition
Opposite
Disposition
Alternate
Opposition
Disposition
Newton
RapsonPSO
TechniqueACO BEE
Algorithm
Fig. 2.2: Types of Different Modulation Methods
Another switching technique in inverter is high Switching Frequency scheme. In this scheme sinusoidal pulse width
modulation (SPWM) technique and space vector (SVPWM) techniques are included. The average load voltage for
every switching interval can be generated by SVPWM method which ensures the suitability of this method in high
voltage applications. Pulse width modulation (PWM) technique has a feature of generating train of pulses relative to
control. They are mostly used in forced commutated inverters serving variable-speed ac motor drive systems.
Whereas I case of sinusoidal PWM (SPWM) technique, the generated control signals are sinusoidal, and the average
generated output voltage will also vary sinusoidally.
Low switching scheme includes fundamental switching frequency scheme namely selective-harmonic elimination
method and space vector control (SVC) method. In SVC method, a voltage vector (Vc) and reference voltage vector
(Vref) are approximately equal so that the space error and designing modulation complexity can be reduced to a
great extent. Hence the errors will be small as compared with the reference vector. The SVC method is more suitable
for large number of voltage level inverters. Selective Harmonic Elimination is implemented with low frequency i.e.
fundamental frequency. It will ensure the elimination of harmonics in the system and maintaining the fundamental
component at desired level. In SHE technique, DC sources decide the number of levels of multilevel inverter which
will again decide the quantity of eliminated harmonics in the system. In SHE technique, calculation of switching
period is done through the preprogrammed angle arrangement fetched from the look-up table. Hence this technique
is more attractive and popular day by day as it contains fundamental switching frequency with less switching losses.
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3. SELECTIVE HARMONIC ELIMINATION
In selective harmonic elimination technique, both fundamental frequency and the switching frequency is equal.
When inverter is switched with fundamental frequency, the switching losses are less and the efficiency of the system
increases. To achieve this, the most important thing is to select proper optimized switching angles at the time of
switching. Consider a case for cascaded H Bridge seven level multilevel inverter as shown in Figure 3.1and whose
output voltage is shown in Figure 3.2.
S1
S4S3
Vdc
S8S7
S6
Vdc
S9
S12S11
S10
Vdc
V
S2
S5
N
Fig. 3.1: H bridge seven level inverter
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Fig. 3.2: Output waveform
For single phase seven level inverter, the output voltage expression is 𝑉(𝑤𝑡) = ∑ 𝑉𝑛 sin(𝑛𝑤𝑡)∞𝑛=1
After fourier series, the above equations can be simplified as 𝑉1 = 4𝑉𝑑𝑐𝜋 [cos(𝛼1) + cos(𝛼2) + cos(𝛼3)] = 𝑉1∗
𝑉5 = 4𝑉𝑑𝑐5𝜋 [cos(5𝛼1) + cos(5𝛼2) + cos(5𝛼3)] = 0
𝑉7 = 4𝑉𝑑𝑐7𝜋 [cos(7𝛼1) + cos(7𝛼2) + cos(7𝛼3)] = 0
These equations are called as nonlinear transcendental equations. From the above equations it is clear that, the first
equation defines the fundamental voltage component which is to be maintained at desired level. Whereas, second
and third equation defines the fifth and seventh harmonic voltage to be reduced to zero. In these equations,
switching angles α1, α2, α3decides the harmonics to be eliminated. The ratio of output fundamental voltage to
maximum output voltage is the modulation index, ma in multilevel inverters. So in terms of modulation index, all the
three nonlinear equations are modified as following.
cos (1 ) cos (2 ) cos (3 ) 3ma
cos (51 ) cos (52 ) cos (53 ) 0
cos (71 ) cos (72 ) cos (73 ) 0
subject to 0
123/ 2
The solution to these nonlinear transcendental equations is determined through optimization methods.
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3.1. Newton Raphson Algorithm
Obtaining solutions to nonlinear transcendental equations is very complicated and time consuming. Hence by
iterative methods also we can get the solutions but the time of convergence may be more. One of the iterative
method is Newton Raphson Algorithm. Only drawback with this method is selection of initial guess which is close
to initial guess. It is implemented for obtaining the solutions with the following steps.
1. Assume any initial random angleα0
2. Fix ma = 0
3.Define objective function F(α0) , B (ma), Jacobian J (α0).
4.Update angle α by small incremental change ∆α
J1 (0 )[( B(ma )F (0 )]
5. Update angles as α(k+1) = α(k) + ∆α(k).
6. Select switching angles in tolerable range.
7. Go to steps (3) to (6) again with fixed iterations.
8.Revise modulation index ‘ma’
9. Repeat steps (2) to (8)
10. End of Algorithm and plot the results.
The switching pattern for single phase H-Bridge cascaded multilevel inverter is shown in Figure 3.3 below. So, the
firing angles are computed offline by N-R method which fed to the switching devices.
)))(cos((cos )1(
1
)1(
kk
abs
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S1
S4S3
Vdc
S8S7
S6
Vdc
S9
S12S11
S10
Vdc
V
S2
S5
N
LOAD
Vdc
α1
α2
α3
By Newton
Raphon Method
Fig. 3.3: Switching Model for inverter
After implementing N-R algorithm different three switching angles are obtained in case of single phase seven level
inverter. Out of three switching angles, one is referred to fundamental voltage and other two angles are referred for
elimination of lower order harmonics i.e. fifth and seventh harmonics in the system. The simulated results obtained
for three different switching angles as shown in Figure 3.4 and Figure 3.5.
0 0.5 1 1.5 2 2.5 30
10
20
30
40
50
60
70
80
90
Modulation Index(M)
Sw
ich
ing
An
gle
s(d
eg
rees)
α1
α2
α3
α1
α3
α2
0 5 10 15 20 25 30 35
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Number of Iterations
Sw
itch
ing
An
gle
s (
Ra
d)
α3
α2
α1
α1
α2
α3
Fig. 3.4: Switching angles Vs ‘ma’ Fig. 3.5: Switching angles Vs iterations
The simulated results are obtained from MATLAB/Simulink environment for single phase 7-Level cascaded H
Bridge inverter. H Bridge subsystem and firing system is shown in Figure 3.6 and Figure 3.7.
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Fig. 3.6: H- Bridge subsystem
The three different switching angles α1=11.50°, α2=28.71°, α3=57.10° are obtained from the iterative N-R algorithm.
Switching pulses for respective switches(IGBT) in H-bridge system is given after comparing these constant values
of switching angles with sine wave. These pulses are now provided to upper and lower switches of every leg of H
bridge inverter without considering the dead band. The switching pulses obtained from simulation are shown in
Figure 3.9.
Fig. 3.7: Firing pulses generation
2
G2
1
G1
<=
Relational
Operator2
>=
Relational
Operator1
A
From1
A
From
2
In2
1
In1
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Fig. 3.8: Simulink Model
In Figure. 3.8, three phase inverter using H-Bridge blocks are shown. All are linked in series to build to form a entire
three phase seven level multilevel inverter. The equivalent switching as shown in Figure 3.7 in MATLAB Simulink
environment. Subsystem shows the total 3 H-Bridges/phase of seven level inverter. The firing pulses to the
respective H-Bridges are supplied through switches S1-S4. After supplying input, a stepped waveform is received at
the output side. Here simulation is done for 7-Level inverter. For more levels, more number of H-Bridge blocks can
be further added.
Fig. 3.9: PWM generation
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1
Time, s
PW
M 1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1
Time, s
PW
M2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1
Time, s
PW
M3
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0 5 10 15 200
20
40
60
80
100
Harmonic order
Fundamental (50Hz) = 32.36 , THD= 4.66%
Mag (
% o
f F
undam
enta
l)
After obtaining switching angles from N-R method, and MATLAB simulation, the phase output voltage and the
harmonics and THD plot is shown in Figure 3.10and in Figure 3.11.
Fig. 3.10: Phase Voltage Fig. 3.11: FFT Plot
From the above simulation results, the phase output voltage is obtained for single phase seven level inverter. For
this configuration 3- H bridges are constructed with 12V DC supply. Hence phase output voltage is obtained as 36V.
But the FFT analysis shows that all the lower order harmonics are not completely eliminated and also the value for
Total Harmonic Distortion (THD) is found to be 4.66%. Hence the main objective of SHE Technique is not
completely satisfied with the only one algorithm. Hence there is a need of investigation of advanced algorithm to
solve nonlinear transcendental equations including minimization of lower order harmonics along with THD
respectively. A hybrid optimization algorithm is proposed in this monograph which is a combination of Newton
Raphson algorithm and Ant Colony algorithm. Here output of ACO algorithm is fed as a input to the N-R algorithm
and results are computed. Hence the dependency of exact initial guess in N-R algorithm is eliminated in Hybrid
Optimization algorithm.
4. HYBRID OPTIMIZATION ALGORITHM
In hybrid optimization algorithm, combination of ACO and N-R algorithm is implemented and results are obtained
for switching angles. The following steps are to be followed for hybrid optimization algorithm.
Step 1 : Define Objective Function
Step 2 : Define constraints
Step 3 : Find minimum best value
Step 4 :Initially run ACO algorithm for 4iterations
Step 5 :Feed output from ACO algorithm as a input for N-R algorithm.
Step 6 : Run N-R algorithm for convergence.
Step 7 :Obtain solution for switching angles
Step 8 :Stop the algorithm
0 0.005 0.01 0.015 0.02 0.025 0.03-40
-20
0
20
40
Time(sec)
Phase V
oltage
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The following Figure 3.11 shows the flowchart for hybrid algorithm.
Initialization
Construct
Solution
Update Trails
Termination
Condition
Print Best
Solution
Empty Tabu List
Yes
No
Start
1
Initial Guess α0,
set mi =0
Calculate F(α0), A(mi), J( α0)
Calculate ))()()(( 00
1 FmAJi
Update α(k+1)=α(k)+∆α(k)
Is
Feasible range of α
Obtained?
NO
YES
))))1((cos((cos)1( 1 kabsk
mi =mi +0.001
Is
2/...0 21 s
Is
0 < mi < 1
YES
NO
Plot
α against mi
END
1
Fig. 3.11: Flowchart for hybrid algorithm
From this flowchart it is clear that the Ant Colony Optimization algorithm is first running for few cycles, then the
output obtained from ACO algorithm is given as a input to the N-R algorithm. Hence dependency of random initial
guess in N-R algorithm is somewhat eliminated with the proposed hybrid algorithm. These nonlinear equations are
now resolved by hybrid algorithm. The different switching angles obtained through proposed hybrid algorithm are
found to be α1 = 14.26˚, α2 = 28.34˚, α3 = 41.15˚.These angles are obtained for modulation index ‘ma’ = 0.8. The
solution set obtained after implementing hybrid algorithm provides the optimized values for switching angles. The
different solution sets are obtained after implementing the algorithm as shown in Table 1. Only one solution set is to
be selected at a time for simulation purpose which will solve the objective of retaining fundamental component at
the desired level and reducing the lower order harmonics and THD simultaneously. Also by using these switching
angles the lower order harmonics particularly odd harmonics i.e. fifth and seventh harmonics are comparatively very
very less and can be eliminated easily. Also the Total Harmonic Distortion (THD) ratio is reduced greatly.
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Table 1: Switching angles with ma
M.I., ma α1 α2 α3
0.61 5.71˚ 15.55˚ 21.61˚
0.71 24.20˚ 23.56˚ 47.42˚
0.75 18.49˚ 22.52˚ 58.60˚
0.8 14.26˚ 28.34˚ 41.15˚
0.81 19.12˚ 13.82˚ 41.15˚
0.91 25.55˚ 27.70˚ 67.18˚
0.95 23.03˚ 37.34˚ 59.28˚
1.10 21.00˚ 39.34˚ 54.09˚
1.5 28.89˚ 51.56˚ 60.60˚
These obtained switching angles are plotted with respect to modulation index ‘ma’ and as shown in Figure 3.12.
Also the objective function is computed as minimum and shown in Figure 3.13. The phase output voltage of single
phase seven level inverter is shown in Figure 3.14 and the harmonic profile of the system is shown in Figure 3.15.
0 0.2 0.4 0.6 0.8 10
10
20
30
40
50
60
70
80
Modulation Index(M)
Sw
ich
ing A
ng
les(
deg
rees)
α1
α2
α3 α1
α2
α3
90
Fig. 3.12: Variation of Switching Angles Fig. 3.13: Variation of Objective Function
10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
Number of Iteration
Obje
cti
ve F
uncti
on
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Fig. 3.14: Phase Output Voltage Fig. 3.15: FFT Plot
From the all above results, it is clear that the proposed hybrid algorithm is proved to be the effective algorithm for
eliminating lower order harmonics and maintaining the fundamental component at the desired level. The results
show the output voltage of single phase 7-level inverter is containing negligible harmonics with the desired output
phase voltage. Also FFT analysis shows the fifth harmonic is found to be very less as h5 = 0.02% and seventh
harmonic is found to be h7 = 0.01%. Also from the harmonic spectra the Total harmonic distortion (THD) is obtained
as very less i.e. THD = 2.66% as compared to other iterative method. Also the harmonics are in accordance with
IEEE 519 standard i.e. they are within tolerable range.
5. CONCLUSION
Multilevel inverters are proved to be the most significant device in medium power and high power applications. As
far as renewable energy sources are concerned, DC to AC conversion is necessary. So multilevel inverters are
generally satisfying this requirement which ensures the high efficiency. For obtaining sinusoidal output waveform,
modulation techniques are implemented. But all modulation techniques cannot solve the issue of selective harmonic
elimination. The high frequency modulation techniques which include generally sinusoidal pulse width modulation
(SPWM) technique cannot improve the harmonic profile and gives the more switching losses. Many conventional
algorithms cannot find all the solutions to SHE equations. Hence Solution to SHE equations are given by Hybrid
algorithm which is investigated in this monograph.
Hybrid Algorithm is applied for obtaining optimized values of switching angles required for switching the
power devices of single phase seven level multilevel inverter. Simulation of this cascaded H- Bridge inverter is done
in MATLAB/ Simulink environment. This simulation is presented as a systematic approach for single phase seven
level multilevel inverter. All the switching angles are computed for modulation index 0.8 and desired values are
obtained. Owing to this proposed algorithm, all the lower order harmonics are eliminated easily and the desired
0 0.005 0.01 0.015 0.02 0.025 0.03-40
-20
0
20
40
Time(sec)
Ph
ase V
oltag
e
0 5 10 15 20 25 30 35 400
20
40
60
80
100
Harmonic order
Fundamental (50Hz) = 31.29 , THD= 2.66%
Mag (
% o
f F
undam
enta
l)
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fundamental voltage is obtained its value. All the simulated results show the effectiveness of hybrid algorithm.
Convergence and finding solutions is very rapid in case of hybrid algorithm.
6. FUTURE SCOPE
In this monograph, analysis of single phase multilevel inverter is carried out with equal DC sources. Further
extension of this work can be extended with asymmetrical configuration. Also this work is further extended to three
phase multilevel inverter with separate input i.e. separate DC sources and solution to SHE equations can be found
out by hybrid algorithm. Here only fifth and seventh harmonics are considered along with THD for elimination.
Further study can be extended to eliminate different order harmonics with different modulation indices.
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7. REFERNCES
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[3] F. Z. Peng, J. S. Lai, J. McKeever, and J. VanCoevering, “A multilevel voltage-source converter system with balanced DC voltages,” In Power Electron. Specialists Conf., PESC'95 Record., 26th Annual IEEE, vol. 2, pp. 1144-1150, 1995.
[4] C. Newton, and M. medium/high-voltage pp. 21-26, 1998.“Multi-level convertors a real solution to drives?,” Power Engineering Journal, vol. 12, no. 1,
[5] L. M. Tolbert, F. Z. Peng, and T. G. Habetler, “Multilevel inverters for electric vehicle applications,” In Power Electron. in Transportation, pp. 79-84, 1998.
[6] J. Rodriguez, S. Bernet, P. K. Steimer, and I. E. Lizama, “A survey on neutral-point-clamped inverters,” IEEE Trans. on Ind. Electron., vol. 57, no. 7, pp. 2219-2230, 2010.
[7] X. Yuan, and Ivo Barbi, “Fundamentals of a new diode clamping multilevel inverter,” IEEE Trans. on power Electron., vol. 15, no. 4, pp. 711-718, 2007.
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